Expression For The Excess Pressure Due To Water Hammer
When a valve with a volumetric flow rate Q is closed, an excess pressure δP is created upstream of the valve, whose value is given by the Joukowsky equation:
In this expression:
- overpressurization δP is expressed in Pa;
- Q is the volumetric flow in m3/s;
- Zh is the hydraulic impedance, expressed in kg/m4/s.
The hydraulic impedance Zh of the pipeline determines the magnitude of the water hammer pulse. It is itself defined by:
with:
- ρ the density of the liquid, expressed in kg/m3;
- A cross sectional area of the pipe, m2;
- Beff effective modulus of compressibility of the liquid in the pipe, expressed in Pa.
The latter follows from a series of hydraulic concepts:
- compressibility of the liquid, defined by its adiabatic compressibility modulus Bl, resulting from the equation of state of the liquid generally available from thermodynamic tables;
- the elasticity of the walls of the pipe, which defines a modulus of equivalent compressibility Beq. In the case of a pipe of circular cross section whose thickness e is small compared to the diameter D, the equivalent modulus of compressibility is given by the following formula: ; in which E is the Young's modulus (in Pa) of the material of the pipe;
- possibly compressibility Bg of gas dissolved in the liquid, defined by:
- γ being the ratio of specific heats of the gas
- α the rate of ventilation (the volume fraction of undissolved gas)
- and P the pressure (in Pa).
Thus, the effective compressibility modulus is:
As a result, we see that we can reduce the water hammer by:
- increasing the pipe diameter at constant flow, which reduces the inertia of the liquid column;
- choosing to use a material with a reduced Young's modulus;
- introducing a device that increases the flexibility of the entire hydraulic system, such as a hydraulic accumulator;
- where possible, increasing the percentage of undissolved air in the liquid.
Read more about this topic: Water Hammer
Famous quotes containing the words expression, excess, pressure, due, water and/or hammer:
“The fact is, the public make use of the classics of a country as a means of checking the progress of Art. They degrade the classics into authorities. They use them as bludgeons for preventing the free expression of Beauty in new forms.”
—Oscar Wilde (1854–1900)
“There is an excess both in happiness and misery above our power of sensation.”
—François, Duc De La Rochefoucauld (1613–1680)
“Destiny is an absolutely definite and inexorable ruler. Physical ability and moral determination count for nothing. It is impossible to perform the simplest act when the gods say “no.” I have no idea how they bring pressure to bear on such occasions; I only know that it is irresistible.”
—Aleister Crowley (1875–1947)
“Westerners inherit
A design for living
Deeper into matter—
Not without due patter
Of a great misgiving.”
—Robert Frost (1874–1963)
“It would not be an easy thing to bring the water all the way to the plain. They would have to organize a great coumbite with all the peasants and the water would unite them once again, its fresh breath would clear away the fetid stink of anger and hatred; the brotherly community would be reborn with new plants, the fields filled with to bursting with fruits and grains, the earth gorged with life, simple and fertile.”
—Jacques Roumain (1907–1945)
“This Administration has declared unconditional war on poverty and I have come here this morning to ask all of you to enlist as volunteers. Members of all parties are welcome to our tent. Members of all races ought to be there. Members of all religions should come and help us now to strike the hammer of truth against the anvil of public opinion again and again until the ears of this Nation are open, until the hearts of this Nation are touched, and until the conscience of America is awakened.”
—Lyndon Baines Johnson (1908–1973)